Decomposition Polyhedra of Piecewise Linear Functions
Marie-Charlotte Brandenburg, Moritz Grillo, Christoph Hertrich
TL;DR
This paper investigates how to efficiently decompose continuous piecewise linear functions into minimal convex components, with implications for optimization, neural networks, and submodular functions, by analyzing the structure of their decompositions.
Contribution
It introduces a polyhedral framework for understanding CPWL function decompositions, disproves a recent approach, and explores minimal and unique decompositions with applications in neural networks and submodular functions.
Findings
Decomposition sets form intersections of translated cones.
Irreducible decompositions correspond to bounded faces of a polyhedron.
Identifies cases with unique minimal decompositions.
Abstract
In this paper we contribute to the frequently studied question of how to decompose a continuous piecewise linear (CPWL) function into a difference of two convex CPWL functions. Every CPWL function has infinitely many such decompositions, but for applications in optimization and neural network theory, it is crucial to find decompositions with as few linear pieces as possible. This is a highly challenging problem, as we further demonstrate by disproving a recently proposed approach by Tran and Wang [Minimal representations of tropical rational functions. Algebraic Statistics, 15(1):27-59, 2024]. To make the problem more tractable, we propose to fix an underlying polyhedral complex determining the possible locus of nonlinearity. Under this assumption, we prove that the set of decompositions forms a polyhedron that arises as intersection of two translated cones. We prove that irreducible…
Peer Reviews
Decision·ICLR 2025 Spotlight
The problem of decomposing CPWL functions as the difference of convex CPWL functions is interesting. No finite procedure is currently known for finding a minimal decomposition. This work provides a new perspective, based on polyhedral geometry, that guarantees finite convergence, but in the special case where the factors have a fixed supporting polyhedral decomposition
The assumption that the DC decomposition should be with respect to a fixed polyhedral complex seems restrictive and perhaps unmotivated I feel that the paper is hard to read for non-experts in the area (e.g. many technical definitions with not many accompanying figures to help the reader-there are few, but mostly in the Appendix). The main result in Section 6 (Corollary 6.4) for representing CPWL functions as NNs, is only applicable to CPWL functions that are compatible with a regular polyhed
The DC representation of a general CPWL function is an old but fundamental problem with many applications in various engineering fields. This paper proposes an interesting perspective on how to understand and compute the DC components from a given CPWL function. Although I did not have time to check all the proofs in detail, the paper is generally well written, and the results are interesting. In particular, I appreciate the idea of fixing the underlying pieces and the clean characterization of
While the pieces are assumed to be fixed in advance (which is, of course, a limitation, as also explicitly noted by the authors), I believe this work has great potential to motivate further investigation into both the theoretical and algorithmic aspects of the decomposition problem. My comments are as follows: * L200, in the definition of $\mathcal{P}_f^n$, I don't think the set {$x:g_i(x)=\max_j g_j(x)$} must be full dimensional. This may depend on the representation of $f$ given in L199. Als
1. The paper presents an innovative approach by linking decomposition problems with polyhedral geometry, leading to the concept of decomposition polyhedra. This is a very novel idea and may inspire many interesting future works. 2. The theoretical analysis of the paper is very solid, providing us with a deep understanding of CPWL decomposition problem. 3. The paper is well-written and well-organized, clearly stating the main contributions and their applications. 4. The applications to ReLU
The main weakness of the paper is that as stated in Limitations section of the paper. The paper mainly focuses on the development of theories, but does not provide practical implementations and applications of their results.
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Taxonomy
TopicsMathematics and Applications · Computational Geometry and Mesh Generation · Advanced Theoretical and Applied Studies in Material Sciences and Geometry
MethodsSparse Evolutionary Training